摘要
The purpose of the paper is to study sharp weak-type bounds for functions of bounded mean oscillation. Let 0 〈 p 〈 ∞ be a fixed number and let I be an interval contained in R. The author shows that for any φ ： I → R and any subset E I of positive measure, ｜I｜^-1/p/｜E｜1-1/p∫E｜φ -1/｜I｜∫Iφdy｜dx≤｜｜φ｜｜BMO（I）,0〈p≤2,｜I｜^-1/p/｜E｜1-1/p∫E｜φ -1/｜I｜∫Iφdy｜dx≤p/2^2/pe2/p-1｜｜φ｜｜BMO（I）p≥2. For each p, the constant on the right-hand side is the best possible. The proof rests on the explicit evaluation of the associated Bellman function. The result is a complement of the earlier works of Slavin, Vasyunin and Volberg concerning weak-type, L ^p and exponential bounds for the BMO class.
基金
supported by the NCN grant DEC-2012/05/B/ST1/00412
